Wine is fined using either isinglass (fish bladder protein), casein (milk protein) or egg albumen (egg white). All of these products exist as by-products of another industry (ie fish aren't killed just to fine wine with!) They are the best fining agents for ensuring a clear wine. There is no actual fish, milk or egg product left in the wine. It is merely the method for removing anything which may make the wine slightly cloudy or taste off, eg clumps of proteins or yeasts.
Vegan wines use organic materials to do this, eg carbon, clays, plant proteins. There is more work involved in this, which is ultimately reflected in the price, and is why many of the big name brands (at the cheaper end of the market) don't use them.
Re sulphites - SO2 is a natural by-product of wine production (fermentation) so all wines contain some sulphites. If the winemaker adds any SO2 to help preserve the wine, it has to be put on the label ("contains sulphites") when it's over a certain amount. Red wines keep longer naturally, because of the tannins, so contain less added SO2.
The levels of sulphites in any wines is tiny, about 10 times less than the amount in dried fruit! Very very few people are actually allergic to SO2, but one's sensitivity to it can build up as you get older. Buying wine that doesn't say "contains sulphites" just means that there is a small quantity, or no, added SO2, because the naturally-occurring sulphites are felt to be sufficient in that particular blend. The winemaker only adds SO2 to ensure the wine stays fresh in the bottle for more than six months.
Someone also mentioned yeast. Yeasts occur naturally in grapes and it is these which help the natural grape sugars turn to alcohol during fermentation. The yeasts die off during the alcohol production so there wont be any in your wine.
(I work in the wine trade and get asked about this stuff a lot )